On 2015-01-25 22:07, John Cremona wrote:
I would say that this is undefined since the ideal generated by x+2
and x+4 is (2,x) which is not principal. I think that the most useful
definition of gcd is a generator of the ideal generated by the two
objects *if* that ideal is principal.
I disagree. There are rings where a gcd can be defined(*) even if the
ideal is not principal. Multi-variate polynomial rings are a very
natural example. I see no reason to disallow gcd() there.
xgcd() is a different story, there you really need a stronger condition
on the ring, namely you need a Bézout domain (every finitely generated
ideal is principal, which is slightly weaker than PID).
So I think that gcd() should stay as it is and xgcd() should raise
ArithmeticError if the ideal is not principal.
(*) literally as greatest common divisor, where "greatest" is taken
using the partial order defined by division.
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